THEORY OF THE MOTION OF EOCIETS. lg5 



tt a line parallel to the direction in which that force it 

 exerted in the same time that it would have done by the 

 single action of its own impelling: force. Therefore, in or- 

 der to determine the curve which the rocket will describe 

 under the true circumstance of its motion, let us suppose 

 that if gravity had not acted, the rocket would have arrived 

 at C, Pi. V. fig. 1, in the line of its first direction A C in 

 he time t. Then in this case we shall have (Prop. 1.) the 



b c 



tpace described in the time ^, namely A C zz x zz ■ x 



Sam 



Or multiplying by -- — (zz suppose Q) and calling the 



coefficients of the several terms of the series A, B, C, &c. ; 

 it will be Q x iz i* + A <^ + B i* + C f » + D <• + 



&c. ; which reverted into a series of jc is I ~ Q ^ * 



L^H^. Q ,? 



8 



. Q ^ x* 4- &c.) ; the time of describ- 

 ing the distance x along A C from the commencement of 

 motion. 



+ &c. -qi X (x^^ 7-Q' * + — i~^- ^ *' + 



3 A B — 2 A* — C 



Now C D (y) being the distance descended by gravity in 

 the same time ; we therefore get ^ V y (omitting the ^t) ^or 

 the time of the rocket's describing C D by the force of 



gravity : and consequently \ \/ y zzQ* x ( ^* ^. 0= 



5A*— 4B^^^x 

 X + Q x'' 4- &c. J 



Hence, knowing the equation which subsists between 

 AC and C D the track which the rocket describes may bfr 

 drawn ; for it will only be necessary to give some value to 

 X in order to determine the corresponding value of y, and 

 to lay off this upon C D drawn perpendicular to A B, 



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